A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN

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1 A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN JORDAN GOBLET Abstract. A Q k (R n )-valued function is essentially a rule assigning k unordered and non necessarily distinct elements of R n to each element of its domain set A R m. For a Q k (R)-valued function f we construct a decomposition into k branches that naturally inherit the regularity properties of f. Next we prove that a measurable Q k (R n )-valued function admits a decomposition into k measurable branches. An example of a Lipschitzian Q 2 (R 2 )-valued function that does not admit a continuous decomposition is also provided and we state a selection result about multiple-valued functions defined on intervals. We finally give a new proof of Rademacher s Theorem for multiple-valued functions. This proof is mainly based on the decomposition theory and it does not use the Almgren s bi-lipschitzian correspondence between Q k (R n ) and a cone Q R P (n)k. 1. Introduction In his big regularity paper [1], F. J. Almgren introduced the machinery of multiple-valued functions to study the partial regularity of area-minimizing integral currents. He proved that any m-dimensional mass-minimizing integral current is regular except on a set of Hausdorff dimension at most m 2. His regularity theory relies on a scheme of approximation of Dirichlet-minimizing multiple-valued functions. The success of Almgren s regularity theory raises the need of further studying multiple-valued functions and of making his work more accessible. A multiple-valued function f : A R m Q k (R n ) is essentially a rule assigning k unordered and non necessarily distinct elements of R n to each element of its domain. Such maps are studied in complex analysis (see appendix 5 in [8]). Indeed in complex function theory one often speaks of the two-valued function f(z) = z 1/2. This is better considered as a function from R 2 to Q 2 (R 2 ). In Chapter 1 of [1], Almgren proved that the metric space Q k (R n ) is in explicit bi-lipschitzian correspondence with a finite polyhedral cone Q included in a higher dimensional Euclidean space and his analysis is mainly based on this correspondence. In the present paper, we put this correspondence aside and we approach multiple-valued functions by selection arguments. For a Q k (R n )- valued function f, we construct a decomposition into k particular branches f 1,..., f k : R m R n and we study the properties of the branches such as continuity in accordance with those of f. For n = 1, we prove in Proposition 4.1 that f admits Lipschitzian [resp. Hölder continuous, continuous, measurable] branches if f is Lipschitzian [resp. Hölder continuous, continuous, measurable]. For n > 1, we show in Proposition 5.1 that f can be split into measurable branches if f is measurable and we complete this result by a Lusin type Theorem for multiple-valued functions. We also provide an example of a Lipschitzian Q 2 (R 2 )-valued function that does not admit a continuous decomposition and state a selection result already proved in [1] and [3] about multiple-valued functions defined on closed intervals. We finally recall what is meant by differentiability in the context of multiple-valued functions. Essentially a multiple-valued function f is said to be affinely approximatable at a R m if it is possible to approach f near a by a multiple-valued function admitting an affine decomposition. We suggest an original proof of Rademacher s Theorem based on the selection theory : Date: 30th December Mathematics Subject Classification. 54C65. Key words and phrases. Multiple-valued functions, selection, Rademacher s Theorem. 1

2 2 JORDAN GOBLET Theorem. Let f : R m Q k (R n ) be a Lipschitzian multiple-valued function. Then f is strongly affinely approximatable at L m almost all points of R m. We close our paper by using the 1-dimensional selection result to show that these affine approximations control the variation of f: Theorem. Let f : R m Q k (R n ) be a Lipschitzian multiple-valued function and [a, b] := {a + t(b a) t [0, 1]} where a, b R m such that a b. If f is affinely approximatable at H 1 almost all points of [a, b] then F(f(a), f(b)) Af(x) dh 1 (x). [a,b] 2. Preliminaries The scalar product of two vectors x, y R n is denoted (x y), and the Euclidean norm of x R n is denoted x. For a fixed space X with a metric d, let B(a, r) = {x X d(a, x) r}, U(a, r) = {x X d(a, x) < r} be the closed and open balls with center a X and radius r > 0. If L : R m R n is a linear mapping, we set L = sup{ L(x) x R m with x 1}. For each point x i R n, x i denotes the Dirac measure at x i. Denote Q k (R n ) = { k x i x i R n }, where x i and x j are not necessarily distinct for i j and the integer k is already fixed. We can define a topology on Q k (R n ) by the equivalent metrics G and F: G x i, ) { ( k k j=1 y j = min F x i y σ(i) 2 ) 1/2 σ is a permutation of {1,..., k} }, x i, ) { k j=1 y k } j = min x i y σ(i) σ is a permutation of {1,..., k}. We easily prove that (Q k (R n ), G) is a complete separable metric space. A map f : A R m Q k (R n ) will be called a multiple-valued function or a Q k (R n )-valued function. Let us note that a Q 1 (R n )-valued function is nothing else than a classical function since Q 1 (R n ) is in bijection with R n. To begin with, let us pay attention to a problem that will make it possible for us to reduce significantly the size of the coming proofs. Considering f 1,..., f k : A R m R n, we can define the following Q k (R n )-valued function (1) f := f i and wonder if f inherits the regularity properties of f 1,..., f k. We can then easily check Proposition 2.1. Proposition 2.1. Let f 1,..., f k : A R m R n and f defined by (1). The following assertions are checked: (1) If f 1,..., f k are Lipschitzian on B A then f is Lipschitzian on B and Lip (f) Lip2 (f i )) 1/2. (2) If f 1,..., f k are continuous on B A then f is continuous on B. To end this section, we will define what is meant by measurable Q k (R n )-valued function and prove a proposition which is similar to Proposition 2.1. As in [4] and [5], we call measure what is usually called outer measure.

3 A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN 3 Definition 2.1. Let µ be a measure on R m. A multiple-valued function f : A R m Q k (R n ) is said µ-measurable if for all open G Q k (R n ), f 1 (G) is µ-measurable. Proposition 2.2. Let f 1,..., f k : A R m R n and f defined by (1). If f 1,..., f k are µ- measurable then f is µ-measurable. Proof. Since Q k (R n ) is separable it is enough to show that f 1 (U) is µ-measurable for all open balls U Q k (R n ). Let v = k x i Q k (R n ) and r > 0. We note that { ( f 1 (U(v, r)) = y A G v, ) } k f i(y) < r so f is µ-measurable. = σ = σ { y A k x i f σ(i) (y) 2 < r 2 } r 1,..., r k rational r r2 k < r2 k f 1 σ(i) (B(x i, r i )) 3. Canonical Decomposition The central idea in this section consists in decomposing multiple-valued functions in the most efficient possible way. If one has a Q k (R n )-valued function defined on A R m, it is easy to find maps f 1,..., f k : A R n such that f = k f i on A. However we will be concerned with the existence of branches which preserve some properties of f such as measurability, continuity... We start by defining some necessary tools to this study. For all n, k N 0, we define the functions β (n,k),1 : Q k (R n ) R n : v min (spt(v)) where R n is endowed with a total order. For k N 0 and v Q k (R n ) fixed, we will define by induction with respect to i {1,..., k} a sequence of pairs (v i, β (n,k),i (v)) where v i Q k i+1 (R n ) and β (n,k),i (v) R n. Let us assume that v 1 = v and that the sequence is defined until the index i, we then set successively v i+1 := v i β (n,k),i (v) and β (n,k),i+1 (v) := β (n,k i),1 (v i+1 ). By this process, we have just built the functions (2) β (n,k),i : Q k (R n ) R n for i = 1,..., k associated to a total order on R n and such that (3) v = β (n,k),i (v) for all v Q k (R n ). Consequently, if we consider a multiple-valued function f : A R m Q k (R n ), an immediate consequence of (3) is that f = k f i with (4) f i := β (n,k),i f for i = 1,..., k. Subsequently, we will study the properties of the selection defined by (4), assuming that R n is endowed with the lexicographical order. 4. Decomposition properties for n = 1 Lemma 4.1. For all k N 0, Lip ( β (1,k),i ) 1 for i = 1,..., k.

4 4 JORDAN GOBLET Proof. Let k N 0 and v = k j=1 x j, w = k j=1 y j Q k (R) such that x 1... x k and y 1... y k. By the first part of the proof of Theorem 1.2 in [1], it is clear that for i = 1,..., k. G 2 (v, w) = x j y j 2 x i y i 2 = β (1,k),i (v) β (1,k),i (w) 2 j=1 We can then state Proposition 4.1 which immediately ensues from (4) and Lemma 4.1. Proposition 4.1. Let f : A R m Q k (R) be a multiple-valued function. Then there exist f 1,..., f k : A R m R such that f = k f i on A and the following assertions hold: (1) If f is continuous and ω f,x denote the modulus of continuity of f at x i.e. ω f,x (δ) = sup { G(f(x), f(y)) y A and x y δ} then f 1,..., f k are continuous and ω fi,x ω f,x for i = 1,..., k and for all x A where ω fi,x denotes the modulus of continuity of f i at x i.e. ω fi,x(δ) = sup { f i (x) f i (y) y A and x y δ}. (2) If f is µ-measurable then f 1,..., f k are µ-measurable. This last proposition is useful as it can help us to easily prove a few theorems that are not obvious at first sight. Nevertheless, this proposition only settles when n = 1. Can we hope the same for n > 1? The next section will attempt to answer this question. In order to illustrate the interest of Proposition 4.1, we mention two extension theorems for Q k (R)-valued functions. Theorem 4.1. If A R m is closed and f : A Q k (R) is continuous, then f has a continuous extension f : R m Q k (R). Proof. This follows from Proposition 4.1, Tietze s extension Theorem and Proposition 2.1. Theorem 4.2. If A R m and f : A Q k (R) is Lipschitzian, then f has a Lipschitzian extension f : R m Q k (R) with Lip ( f) k 1/2 Lip (f). Proof. This follows from Proposition 4.1, Kirszbraun s extension Theorem and Proposition Decomposition properties for n > 1 As shown earlier, studying the hypothetical properties of the selection defined by (4) boils down to studying the properties of the functions (2). For n = 1, we showed that these maps are Lipschitzian. For n > 1, we will prove that they are merely Borelian. Let e 1,..., e n denote the standard basis vectors of R n. This notation will only be used in Lemma 5.1. Recall that R n is endowed with the lexicographical order. Lemma 5.1. For all k N 0 and n N 0 \{1}, β (n,k),i is Borelian for i = 1,..., k. Proof. Let k N 0 and n N 0 \{1}. We will partition Q k (R n ) into a countable union of closed sets so that the restrictions of the functions β (n,k),i to these closed sets are continuous. It will then be clear that the functions β (n,k),i are Borelian. It is obvious that Q k (R n ) = a=1k a

5 A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN 5 where K a = Q k (R n ) { } x i if (x i e l ) (x j e l ) for l {1,..., n} then (x i e l ) (x j e l ) 1/a. Let us show that the sets K a are closed and that the restrictions of the functions β (n,k),i to these sets are continuous. Fix a N 0. We take a sequence (v q = k x i,q ) q N K a such that v q v as q and we will show that v K a. Let us consider any subsequence of (v q ) q N still denoted by (v q ) q N. Since (v q ) q N converges, it is bounded; hence there exists M > 0 such that G 2 (v q, k 0 ) = k x i,q 2 < M for all q N. Consequently the sequence (x i,q ) q N is bounded for i = 1,..., k. Therefore there exist q 1 < q 2 <... and x 1, x 2,..., x k R n such that for all i {1,..., k}, the subsequence (x i,qr ) r N converges to x i as r. Then the subsequence (v qr ) r N converges to k x i as r ; hence v = k x i. If (x i e l ) (x j e l ) for l {1,..., n} then there exists p N such that (x i,qr e l ) (x j,qr e l ) for all r p hence (x i,qr e l ) (x j,qr e l ) 1/a. By taking the limit, we deduce that (x i e l ) (x j e l ) 1/a; hence v K a. It only remains to prove that the functions β (n,k),i restricted to K a are continuous. For i, j {1,..., k} such that i j, we state the following important remarks: (1) If (x i e l ) = (x j e l ) for l {1,..., n} then there exists p N such that for all r p, we have (x i,qr e l ) = (x j,qr e l ). Otherwise, there exist q r1 < q r2 <... such that (x i,qrs e l ) (x j,qr s e l) for all s N hence (x i,qr s e l) (x j,qr s e l) 1/a which on letting s implies the contradiction (x i e l ) (x j e l ) 1/a. (2) if (x i e l ) < (x j e l ) for l {1,..., n} then there exists p N such that for all r p, we have (x i,qr e l ) < (x j,qr e l ). Otherwise, there exist q r1 < q r2 <... such that (x i,qr s e l) (x j,qrs e l ) for all s N which on letting s implies the contradiction (x i e l ) (x j e l ). (3) if (x i e l ) > (x j e l ) for l {1,..., n} then there exists p N such that for all r p, we have (x i,qr e l ) > (x j,qr e l ). Otherwise, there exist q r1 < q r2 <... such that (x i,qr s e l) (x j,qr s e l) for all s N which on letting s implies the contradiction (x i e l ) (x j e l ). Consequently, if x 1,..., x k are ordered in a certain way then there exists p N such that x 1,qr,..., x k,qr are ordered in the same way for all r p. Therefore β (n,k),i (v qr ) β (n,k),i (v) for i = 1,..., k. The following result is a consequence of (4) and Lemma 5.1. Proposition 5.1. Let µ be a measure on R m and f : A R m Q k (R n ) be a µ-measurable multiple-valued function. Then there exist µ-measurable functions f 1,..., f k : A R m R n such that f = k f i on A. The following theorem is nothing else than an adaptation of Lusin s Theorem to multiple-valued functions. Theorem 5.1. Let µ be a Borel regular measure on R m and f : R m Q k (R n ) be a µ-measurable multiple-valued function. Assume A R m is µ-measurable and µ(a) <. Fix ε > 0. Then there exists a compact set C A and f 1,..., f k : R m R n such that (1) f = k f i on R m, (2) µ(a\c) < ε, (3) f i C is continuous for i = 1,..., k, and (4) f C is continuous.

6 6 JORDAN GOBLET Proof. By Proposition 5.1, there exist µ-measurable functions f 1,..., f k : R m R n such that f = k f i on R m. We then infer from Lusin s Theorem that for i = 1,..., k, there exists a compact set C i A such that µ(a\c i ) < ε k and f i Ci is continuous. We set C := k C i A which is compact. Then it is clear that µ(a\c) < ε and f C is continuous by Proposition 2.1. To end this section, we show that a continuous Q k (R n )-valued function doesn t necessarily admit a continuous decomposition if n > 1. We consider the Lipschitzian Q 2 (R 2 )-valued function f : S 1 R 2 Q 2 (S 1 ) : x = (cos θ, sin θ) f(x) = (cos θ 2, sin θ 2 ) + ( cos θ 2, sin θ 2 ) where S 1 := {(cos θ, sin θ) θ [0, 2π)} is the unit circle centered on the origin, and we study the following selection problem: can one select for all x S 1, a point g(x) spt(f(x)) in such a way that g is continuous on S 1? Assume that such a function exists. We introduce the mapping h : S 1 S 1 : x = (cos θ, sin θ) h(x) = (cos 2θ, sin 2θ) and it is obvious that h is continuous and that its topological degree, written deg(h), is 2. On the other hand, h g = id S 1 so that deg(h g) = 1 and the topological degree of g is necessarily an integer which refutes the equality deg(h g)=deg(h) deg(g). Consequently, a continuous Q k (R n>1 )-valued function doesn t necessarily admit a continuous decomposition. This also proves that there is no order on R n>1 so that the functions defined by (2) are continuous. Nevertheless, we can prove that every continuous multiple-valued function f defined on a closed interval can be split into single branches which inherit the modulus of continuity of f. Proposition 5.2. Let f : [a, b] R Q k (R n ) be a continuous multiple-valued function and ω f denote the modulus of continuity of f. Then there exist continuous functions f 1,..., f k : [a, b] R R n such that f = k f i on [a, b] and ω fi n 1/2 k ω f where ω fi denotes the modulus of continuity of f i for i = 1,..., k. Proof. We will only prove that if f is Lipschitzian then f admits a Lipschitzian selection f 1,..., f k such that Lip(f i ) Lip(f) for i = 1,..., k since it is the unique result we will need later. For a general argument, see the original Proposition 1.10 in [1] or Theorem 1.1 in [3]. We will construct multiple-valued functions f j : [a, b] Q k (R n ) and functions f j 1,..., f j k : [a, b] R n with f j = k f j i for each j = 1, 2,... such that (passing to a subsequence of the j s) one obtains Lipschitzian functions f i = lim j f j i with f = k f i. For fixed j N 0 and for each l = 0, 1,..., j we set t l = a + l t where t = (b a)/j. Then we choose f j : [a, b] Q k (R n ) subject to the requirement that for each l = 1,..., j and t [t l 1, t l ] f j (t) = tl t t p i + t t l 1 q i t corresponding to some choice of p 1,..., p k, q 1,..., q k such that f(t l 1 ) = k p i, f(t l ) = k q i and G(f(t l 1 ), f(t l )) = ( k p i q i 2 ) 1/2. One readily verifies the existence of piecewise affine continuous functions f j 1,..., f j k : [a, b] Rn such that f j = k f j i. We choose and fix such functions for each j. We can then easily prove that f(x) = lim j f j (x) for all x [a, b]. Let us fix j N 0 and u {1,..., k}. We will now prove that fu j is in fact Lipschitzian with Lip(fu) j Lip(f). Suppose a c < d b and r, s {1,..., j} such that c [t r 1, t r ] and d [t s 1, t s ]. We consider two cases.

7 A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN 7 (1) Suppose that r = s. We obtain that f j u(c) f j u(d) ( k t r d t p i + d t r 1 t q i t r c t p i c t r 1 t q i 2 ) 1/2 = c d t p i c d t q i 2) 1/2 = d c t p i q i 2 ) 1/2 = d c t G(f(t r 1 ), f(t r )) Lip(f) (d c). (2) Suppose that r < s. We obtain that f j u(c) f j u(d) f j u(c) f j u(t r ) + f j u(t r ) f j u(t r+1 ) f j u(t s 2 ) f j u(t s 1 ) + f j u(t s 1 ) f j u(d) Lip(f) ((t r c) + (t r+1 t r ) (t s 1 t s 2 ) + (d t s 1 )) = Lip(f) (d c). We conclude that the sequences {f j 1 } j N,..., {f j k } j N belong to the following closed, bounded and equicontinuous family Ω = { y C([a, b], R n ) y(x) y( x) Lip(f) x x for all x, x [a, b] and y(a) max{ v v spt(f(a))} }. { } By Arzela-Ascoli s Theorem, there exist j 1 < j 2 <... and f 1,..., f k Ω such that f j l i l N converges uniformly to f i as l for i = 1,..., k. It is immediate that f = lim l f j l = k lim l f j l i = k f i. 6. Rademacher s Theorem for multiple-valued functions Before presenting a theorem for multiple-valued functions identical to Rademacher s Theorem, we will define the meaning of differentiability and approximative differentiability for multiplevalued functions in the way they were originally expressed by Almgren in [1]. Definition 6.1. A multiple-valued function f : R m Q k (R n ) is said affine if and only if there exist affine maps g 1,..., g k : R m R n such that f = k g i. One can checks that if f = k g i is such an affine function, then the functions g 1,..., g k are uniquely determined up to order. Definition 6.2. Let A R m be an open set and f : A Q k (R n ). One says that f is affinely approximatable at a A if and only if there is an affine multiple-valued function g : R m Q k (R n ) such that for all ε > 0 there exists δ > 0 such that G(f(x), g(x)) ε x a for all x B(a, δ) A. Definition 6.3. Let A R m be an open set and f : A Q k (R n ). One says that f is approximately affinely approximatable at a A if and only if there is an affine multiple-valued function g : R m

8 8 JORDAN GOBLET Q k (R n ) satisfying In other words, for any ε > 0, G(f(x), g(x)) ap lim = 0. x a x a L m (B(a, r) {x G(f(x), g(x)) > ε x a }) lim r 0 L m = 0. (B(a, r)) In case f is affinely approximatable [resp. approximately affinely approximatable] at a, the affine multiple-valued function g is seen to be uniquely determined and is denoted Af(a) [resp. apaf(a)]. We set Af(a) = k g i g i (0), where g 1,..., g k are the unique affine functions such that Af(a) = k g i. Definition 6.4. Let A R m be an open set and f : A Q k (R n ). One says that f is strongly affinely approximatable [resp. strongly approximately affinely approximatable] at a A if and only if f is affinely approximatable [resp. approximately affinely approximatable] at a and if g 1,..., g k : R m R n are affine maps such that Af(a) = g i [resp. apaf(a) = then g i (a) = g j (a) implies g i = g j for all i, j {1,..., k}. g i ], We now state two useful and analogous characterizations for the coming proofs. Proposition 6.1. Let a A R m and f : A Q k (R n ) such that f = k f i where f i : A R n is differentiable at a for i = 1,..., k. Then f is affinely approximatable at a. Proof. For i = 1,..., k and for all x R m, we define g i (x) := f i (a) + Df i (a) (x a) and we set g := k g i which is obviously an affine multiple-valued function. Let ε > 0 and i {1,..., k}. Since f i is differentiable at a, there exists δ i > 0 such that x A and x a δ i implies f i (x) g i (x) ε x a. We then choose δ := min{δ k 1/2 1,..., δ k } > 0 because x A and x a δ implies ( ) 1/2 ( G(f(x), g(x)) f i (x) g i (x) 2 ε 2 k x a 2 ) 1/2 = ε x a. Proposition 6.2. Let a A R m and f : A Q k (R n ) such that f = k f i where f i : A R n is approximately differentiable at a for i = 1,..., k. Then f is approximatively affinely approximatable at a. Proof. For i = 1,..., k and for all x R m, we define (5) g i (x) := f i (a) + apdf i (a) (x a) and we set (6) g := g i which is obviously an affine multiple-valued function. Let ε > 0 and r > 0. We easily check that

9 A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN 9 L m (B(a,r) {x R m G(f(x),g(x))>ε x a }) L m (B(a,r)) = Lm (B(a,r) {x R m G(f(x),g(x)) ε x a } c ) L m (B(a,r)) Lm (B(a,r) {x R m k fi(x) gi(x) 2 ε 2 x a 2 } c ) L m (B(a,r)) Lm (B(a,r) ( k {x R m f i (x) g i (x) 2 ε 2 k 1 x a 2 }) c ) L m (B(a,r)) = Lm (B(a,r) ( k {x R m f i (x) g i (x) 2 >ε 2 k 1 x a 2 })) L m (B(a,r)) and the right-hand side converges to zero as r 0. k L m (B(a,r) {x R m f i (x) g i (x) >εk 1/2 x a }) L m (B(a,r)) Theorem 6.1. Let f : R m Q k (R) be a Lipschitzian multiple-valued function. Then f is strongly affinely approximatable at L m almost all points of R m. Proof. According to Proposition 4.1, there exist Lipschitzian functions f 1,..., f k : R m R such that f = k f i. We then deduce from the classical Rademacher s Theorem that there exists B R m such that L m (B) = 0 and f i is differentiable on B c for i = 1,..., k. Proposition 6.1 implies that f is affinely approximatable on B c with Af(a) = k f i(a) + Df i (a) ( a) for all a B c. For all i, j {1,..., k}, we define Z ij = {a B c f i (a) = f j (a)} and we deduce from Corollary 1 in Section 3.1 of [4] that Df i (a) = Df j (a) for L m almost all a Z ij. The proof of Rademacher s Theorem is no longer easy when n > 1 because a Lipschitzian multiple-valued function does not necessarily admit a Lipschitzian decomposition. The first proof was suggested by Almgren in [1] and is based on the existence of a bi-lipschitzian correspondence between Q k (R n ) and a cone Q R P (n)k where P (n) is an integer depending on n. The following proof doesn t use this correspondence. Lemma 6.1. Let A R m and a A. Suppose that f 1,..., f k : A R n are continuous at a. Then there exists 0 < r(a) such that { } min f i (x) f σ(i) (a) 2 = f i (x) f i (a) 2 x U(a, r(a)) A. σ Proof. We define ε := min { f i (a) f j (a) i, j {1,..., k} and f i (a) f j (a)}. Let i {1,..., k}. By the continuity of f i at a, there exists δ i > 0 such that x A and x a < δ i implies f i (x) f i (a) < ε 2. We choose r(a) := min{δ 1,..., δ k } > 0. Indeed for i {1,..., k}, σ a permutation of {1,..., k} and x U(a, r(a)) A, we obtain the two following cases: (1) if f σ(i) (a) = f i (a) then f i (x) f σ(i) (a) = f i (x) f i (a). (2) if f σ(i) (a) f i (a) then it is clear that f σ(i) (a) / U(f i (a), ε). It ensues that f i (x) f σ(i) (a) f i (x) f i (a), otherwise we have the following contradiction ε f i (a) f σ(i) (a) f i (x) f σ(i) (a) + f i (x) f i (a) < 2 f i (x) f i (a) < ε. Theorem 6.2. Let f : R m Q k (R n ) be a Lipschitzian multiple-valued function and A R m be an L m -measurable set such that L m (A) <. Fix ε > 0. Then there exists a compact set C A such that f is strongly affinely approximatable at L m almost all points of C and L m (A\C) < ε.

10 10 JORDAN GOBLET Proof. f is L m -measurable since f is Lipschitzian. By Theorem 5.1, there exist a compact set C A and f 1,..., f k : R m R n such that f = k f i, L m (A\C) < ε and f i C is continuous for i = 1,..., k. Let a C. By Lemma 6.1, there exists r(a) > 0 such that x U(a, r(a)) C implies k f i(x) f i (a) 2 = G 2 (f(x), f(a)) Consequently we get that Lip 2 (f) x a 2. (7) f i (x) f i (a) Lip(f) x a { } L for i = 1,..., k and for all x U(a, r(a)) C. Let Z := a C lim m (B(a,r) C) r 0 L m (B(a,r)) = 1, so that L m (C\Z) = 0 according to the Lebesgue density Theorem. Notice also that L m (B(a, r) C) L m (B(a, r) Z) L m (B(a, r) (C\Z)) 1 = lim r 0 L m = lim (B(a, r)) r 0 L m + lim (B(a, r)) r 0 L m (B(a, r)) }{{} =0 for all a Z. Let ā Z and i {1,..., k}. Let us set h(x) := f i(x) f i (ā) x ā for all x R m and show that { } L ap lim sup x ā h(x) := inf t R lim m (B(ā,r) {x R m h(x)>t}) r 0 L m (B(ā,r)) = 0 (8) Lip(f). Assume that t > Lip (f) and r < r(ā). Observe that (7) implies ( ) L m {x R m f i(x) f i (ā) x ā > t} B(ā, r) L m Lm (B(ā, r)\c) (B(ā, r)) L m (B(ā, r)). Since ā Z, the right-hand side converges to zero as r 0 so that the statement (8) is proved. For i = 1,..., k, we deduce from Theorem of [5] that f i is approximately differentiable on Z i Z with L m (Z\Z i ) = 0. We fix Z := k Z i so that the functions f i are approximately differentiable on Z and L m (Z\Z ) = 0. Let a Z. Proposition 6.2 implies that f is approximately affinely approximatable at a by the affine multiple-valued function defined by (5) and (6). Let us show that f is in fact affinely approximatable at a by the same affine multiple-valued function. Take note that the following argument is a simple adaptation of Lemma of [5]. Let γ > 0. We choose 0 < β < 1 such that ( ) β 1 + Lip(f) + β + ( k apdf i(a ) 2 ) 1/2 < γ 1 β and we define the set W := {x R m G(f(x), g(x)) β x a }. Since f is approximately affinely approximatable at a, there exists δ > 0 such as, if 0 < r < δ, then L m (B(a, r) W c ) L m (B(a, r)) < β m. Let us fix x U(a, δ(1 β)) and regard r := x a 1 β. It is clear that r < δ and B(x, βr) B(a, r). On the other hand, we remark that B(x, βr) W for we otherwise obtain the following 1 This theorem indicates that an approximate local growth condition on f suffices to guarantee that f is the union of a countable family of Lipschitzian functions.

11 A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN 11 contradiction (βr) m α(m) = L m (B(x, βr)) = L m (B(x, βr) W c ) L m (B(a, r) W c ) < β m L m (B(a, r)) = β m r m α(m). where α(m) is the Lebesgue measure of the unit Euclidean ball in R m. Let z B(x, βr) W. We successively have G(f(x), g(x)) G(f(x), f(z)) + G(f(z), g(z)) + G(g(z), g(x)) ) 1/2 Lip(f) x z + β z a + g i(z) g i (x) 2 ) 1/2 Lip(f)βr + β( z x + x a ) + apdf i(a )(z x) 2 ) 1/2 Lip(f)βr + β(βr + x a ) + z x apdf i(a ) 2 ) 1/2 Lip(f)βr + β(βr + x a ) + βr apdf i(a ) 2 = β x a (1 + Lip(f)+β+( k apdf i(a ) 2 ) 1/2 1 β We thus obtain that G(f(x), g(x)) < γ x a so f is affinely approximatable at a. Thus f is affinely approximatable on Z. For all i, j {1,..., k}, we define Zij = {a Z f i (a) = f j (a)} and we deduce from Theorem 3 in Section 6.1 of [4] that apdf i (a) =apdf j (a) for L m almost all a Z ij. Consequently f is strongly affinely approximatable at L m almost all points of C. ). Theorem 6.3. Let f : R m Q k (R n ) be a Lipschitzian multiple-valued function. Then f is strongly affinely approximatable at L m almost all points of R m. Proof. Let us show that there exist disjoint, compact sets {C i } Rm such that and for each i = 1, 2,... L m (( C i ) c ) = 0 f is strongly affinely approximatable at L m almost all points of C i. For each positive integer n, set B n = B(0, n). By Theorem 6.2, there exists a compact set C 1 B 1 such that L m (B 1 \C 1 ) 1 and f is strongly affinely approximatable L m almost everywhere on C 1. Assume now C 1,..., C n have been constructed, there exists a compact set C n+1 B n+1 \ n C i such that L m (B n+1 \ n+1 C i) 1 n+1 and f is strongly affinely approximatable at Lm almost all points of C n+1. Consequently the starting assertion is confirmed. For all i N 0, we define the set A i := {a C i f is strongly affinely approximatable at a} and note that L m (C i \A i ) = 0. It is clear that L m (( A i) c ) = 0 so f is strongly affinely approximatable at L m almost all points of R m. Theorem 6.4. Let f : R m Q k (R n ) be a Lipschitzian multiple-valued function and [a, b] := {a + t(b a) t [0, 1]} where a, b R m such that a b. If f is affinely approximatable at H 1 almost all points of [a, b] then F(f(a), f(b)) Af(x) dh 1 (x). [a,b]

12 12 JORDAN GOBLET Proof. By a standard argument applied to Proposition 5.2, there exist Lipschitzian functions f 1,..., f k : [a, b] R n such that f [a,b] = k f i. We introduce the map ϕ : [0, 1] [a, b] : t a + t(b a) and the direction u = b a b a. We then obtain F(f(b), f(a)) k f i(b) f i (a) So it remains to prove the following assertion: = k f i(ϕ(1)) f i (ϕ(0)) = k 1 0 (f i ϕ) (t) dt k 1 0 D uf i (ϕ(t)) ϕ (t) dt = k [a,b] D uf i (x) dh 1 (x). Claim. Let x ]a, b[ such that f is affinely approximatable at x and D u f i (x) exists for i = 1,..., k. Then k D uf i (x) Af(x). Since f is affinely approximatable at x, it is clear that Af(x)(x) = f(x) = k f i(x) so that Af(x)( ) = k f i(x) + L i (x)( x) where L 1 (x),..., L k (x) : R m R n are linear maps. Let T = { t R x + tu [a, b]}. For all t T, let σ t be any permutation which attains the following minimum: min f σ(i) (x + tu) f i (x) tl i (x)(u). σ We will now prove that there exists δ > 0 such that f σt(i)(x) = f i (x) for i = 1,..., k and for all t T satisfying t < δ. Fix j {1,..., k}. By contradiction, let (t l ) l N T be a sequence converging to 0 such that f σtl (j)(x) f j (x) for all l N. Using the continuity of every f q satisfying f q (x) f j (x), we have that there exists ε > 0 such that, for l large enough, Hence, for l large enough, f j (x) f σtl (j)(x + t l u) > ε. min σ k f σ(i)(x + t l u) f i (x) t l L i (x)(u) t l f σ tl (j)(x + t l u) f j (x) t l L j (x)(u) t l f σ tl (j)(x + t l u) f j (x) t l ε t l L j(x). t ll j (x)(u) t l min k Consequently, lim σ f σ(i)(x+t l u) f i(x) t l L i(x)(u) l t l = which contradicts the fact that f is affinely approximatable at x by Af(x). Then, there exists δ j > 0 such that f σt (j)(x) = f j (x) if t T and t < δ j. If we set δ = min{δ j j = 1,..., k} then f σt (i)(x) = f i (x) for i = 1,..., k and for all t T such that t < δ. Fix ε > 0. Since f is affinely approximatable at x by Af(x), there exists γ > 0 such that if t T and 0 < t < γ then f σt(i)(x + tu) f i (x) t L i (x)(u) < ε.

13 A SELECTION THEORY FOR MULTIPLE-VALUED FUNCTIONS IN THE SENSE OF ALMGREN 13 On the other hand, if t T and 0 < t < min{δ, γ} then f σt (i)(x + tu) f σt (i)(x) L i (x)(u) t < ε; hence min f σ(i) (x + tu) f σ(i) (x) σ L i (x)(u) t < ε. Consequently, lim t 0 F fi(x+tu) f i(x) t, ) k L i(x)(u) = 0. Finally, by the triangular inequality, we obtain for all t 0 that F( D u f i (x), L i (x)(u) ) F( D u f i (x), + F( (f i (x + tu) f i (x))/t ) (f i (x + tu) f i (x))/t, L i (x)(u) ) where the right-hand side converges to zero as t 0. Therefore, there exists a permutation σ such that D u f i (x) = L σ(i) (x)(u) for i = 1,..., k hence D u f i (x) = L i (x)(u) L i (x) = Af(x). 7. Acknowledgement The author would like to thank the referee for his helpful suggestions, and to acknowledge the fact that this type of questions was suggested to him by his thesis advisor Thierry De Pauw. The credit of the idea that the selection theory could imply a new proof of Rademacher s Theorem goes to him, and this article wouldn t exist without the constantly challenging discussions we had. References [1] F. J. Almgren, Jr., Almgren s big regularity paper. Q-valued functions minimizing Dirichlet s integral and the regularity of area-minimizing rectifiable currents up to codimension 2. With a preface by Jean E. Taylor and Vladimir Scheffer. World Scientific Monograph Series in Mathematics, 1. World Scientific Publishing Co., Inc., River Edge, NJ, [2] F. J. Almgren, Jr., Dirichlet s problem for multiple-valued functions and the regularity of mass minimizing integral currents, Minimal submanifolds and geodesics (Proc. Japan-United States sem., Tokyo, 1977), North- Holland, Amsterdam, New York, (1979), 1-6. [3] C. De Lellis, C. R. Grisanti, P. Tilli, Regular selections for multiple-valued functions, Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, [4] L. C. Evans R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, [5] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, [6] F. Morgan, Geometric Measure Theory: A Beginners Guide, Academic Press, Boston, [7] H. A. Priestley, Introduction to Complex Analysis, Oxford University Press, Oxford, [8] H. Whitney, Complex Analytic Varieties, Addison-Wesley, Reading, Jordan Goblet, Département de mathématique, Université catholique de Louvain, Chemin du cyclotron 2, 1348 Louvain-La-Neuve (Belgium) address: goblet@math.ucl.ac.be